Eigenfunctions Of The Laplacian On A Riemannian Manifold

Author : Steve Zelditch
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 52,6 Mb
Release : 2017-12-12
Category : Eigenfunctions
ISBN : 9781470410377

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Eigenfunctions of the Laplacian on a Riemannian Manifold by Steve Zelditch Pdf

Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow. The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain. The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.

Author : Steve Zelditch
Publisher : Unknown
Page : 410 pages
File Size : 53,9 Mb
Release : 2017
Category : MATHEMATICS
ISBN : 1470443449

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Eigenfunctions of the Laplacian on a Riemannian Manifold by Steve Zelditch Pdf

Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow. The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain. The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research

Author : Steven Rosenberg
Publisher : Cambridge University Press
Page : 190 pages
File Size : 40,9 Mb
Release : 1997-01-09
Category : Mathematics
ISBN : 0521468310

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The Laplacian on a Riemannian Manifold by Steven Rosenberg Pdf

This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Author : Christopher D. Sogge
Publisher : Princeton University Press
Page : 206 pages
File Size : 41,7 Mb
Release : 2014-03-10
Category : Mathematics
ISBN : 9781400850549

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Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188) by Christopher D. Sogge Pdf

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.

Author : Isaac Chavel
Publisher : Academic Press
Page : 379 pages
File Size : 47,5 Mb
Release : 1984-11-07
Category : Mathematics
ISBN : 9780080874340

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Eigenvalues in Riemannian Geometry by Isaac Chavel Pdf

The basic goals of the book are: (i) to introduce the subject to those interested in discovering it, (ii) to coherently present a number of basic techniques and results, currently used in the subject, to those working in it, and (iii) to present some of the results that are attractive in their own right, and which lend themselves to a presentation not overburdened with technical machinery.

Author : Urakawa Hajime
Publisher : World Scientific
Page : 312 pages
File Size : 40,9 Mb
Release : 2017-06-02
Category : Mathematics
ISBN : 9789813109100

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Spectral Geometry Of The Laplacian: Spectral Analysis And Differential Geometry Of The Laplacian by Urakawa Hajime Pdf

The totality of the eigenvalues of the Laplacian of a compact Riemannian manifold is called the spectrum. We describe how the spectrum determines a Riemannian manifold. The continuity of the eigenvalue of the Laplacian, Cheeger and Yau's estimate of the first eigenvalue, the Lichnerowicz–Obata's theorem on the first eigenvalue, the Cheng's estimates of the kth eigenvalues, and Payne–Pólya–Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdière, that is, the spectrum determines the totality of all the lengths of closed geodesics is described. We give the V Guillemin and D Kazhdan's theorem which determines the Riemannian manifold of negative curvature.

Author : M.-E. Craioveanu,Mircea Puta,Themistocles RASSIAS
Publisher : Springer Science & Business Media
Page : 447 pages
File Size : 42,7 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401724753

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Old and New Aspects in Spectral Geometry by M.-E. Craioveanu,Mircea Puta,Themistocles RASSIAS Pdf

It is known that to any Riemannian manifold (M, g ) , with or without boundary, one can associate certain fundamental objects. Among them are the Laplace-Beltrami opera tor and the Hodge-de Rham operators, which are natural [that is, they commute with the isometries of (M,g)], elliptic, self-adjoint second order differential operators acting on the space of real valued smooth functions on M and the spaces of smooth differential forms on M, respectively. If M is closed, the spectrum of each such operator is an infinite divergent sequence of real numbers, each eigenvalue being repeated according to its finite multiplicity. Spectral Geometry is concerned with the spectra of these operators, also the extent to which these spectra determine the geometry of (M, g) and the topology of M. This problem has been translated by several authors (most notably M. Kac). into the col loquial question "Can one hear the shape of a manifold?" because of its analogy with the wave equation. This terminology was inspired from earlier results of H. Weyl. It is known that the above spectra cannot completely determine either the geometry of (M , g) or the topology of M. For instance, there are examples of pairs of closed Riemannian manifolds with the same spectra corresponding to the Laplace-Beltrami operators, but which differ substantially in their geometry and which are even not homotopically equiva lent.

Author : Isaac Chavel
Publisher : Cambridge University Press
Page : 402 pages
File Size : 40,5 Mb
Release : 1995-01-27
Category : Mathematics
ISBN : 0521485789

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Riemannian Geometry by Isaac Chavel Pdf

This book provides an introduction to Riemannian geometry, the geometry of curved spaces. Its main theme is the effect of the curvature of these spaces on the usual notions of geometry, angles, lengths, areas, and volumes, and those new notions and ideas motivated by curvature itself. Isoperimetric inequalities--the interplay of curvature with volume of sets and the areas of their boundaries--is reviewed along with other specialized classical topics. A number of completely new themes are created by curvature: they include local versus global geometric properties, that is, the interaction of microscopic behavior of the geometry with the macroscopic structure of the space. Also featured is an ambitious "Notes and Exercises" section for each chapter that will develop and enrich the reader's appetite and appreciation for the subject.

Author : Sridhar Mahadevan
Publisher : Now Publishers Inc
Page : 185 pages
File Size : 47,6 Mb
Release : 2009
Category : Computers
ISBN : 9781601982384

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Learning Representation and Control in Markov Decision Processes by Sridhar Mahadevan Pdf

Provides a comprehensive survey of techniques to automatically construct basis functions or features for value function approximation in Markov decision processes and reinforcement learning.

Author : Pierre H. Berard
Publisher : Springer
Page : 284 pages
File Size : 41,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540409588

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Spectral Geometry by Pierre H. Berard Pdf

Author : Wolfgang Arendt,Wolfgang P. Schleich
Publisher : John Wiley & Sons
Page : 502 pages
File Size : 49,6 Mb
Release : 2009-07-10
Category : Science
ISBN : 9783527628032

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Mathematical Analysis of Evolution, Information, and Complexity by Wolfgang Arendt,Wolfgang P. Schleich Pdf

Mathematical Analysis of Evolution, Information, and Complexity deals with the analysis of evolution, information and complexity. The time evolution of systems or processes is a central question in science, this text covers a broad range of problems including diffusion processes, neuronal networks, quantum theory and cosmology. Bringing together a wide collection of research in mathematics, information theory, physics and other scientific and technical areas, this new title offers elementary and thus easily accessible introductions to the various fields of research addressed in the book.

Author : Thierry Aubin
Publisher : American Mathematical Soc.
Page : 198 pages
File Size : 42,8 Mb
Release : 2001
Category : Geometry, Differential
ISBN : 9780821827093

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A Course in Differential Geometry by Thierry Aubin Pdf

This textbook for second-year graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. Chapter I explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. Chapter II deals with vector fields and differential forms. Chapter III addresses integration of vector fields and p-plane fields. Chapter IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative. Chapter V specializes on Riemannian manifolds by deducing global properties from local properties of curvature, the final goal being to determine the manifold completely. Chapter VI explores some problems in PDEs suggested by the geometry of manifolds. The author is well-known for his significant contributions to the field of geometry and PDEs - particularly for his work on the Yamabe problem - and for his expository accounts on the subject. The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory.

Author : Jeff Cheeger,David G. Ebin
Publisher : Newnes
Page : 183 pages
File Size : 45,6 Mb
Release : 2009-01-15
Category : Computers
ISBN : 9780444107640

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Comparison Theorems in Riemannian Geometry by Jeff Cheeger,David G. Ebin Pdf

Comparison Theorems in Riemannian Geometry

Author : Thomas Friedrich
Publisher : American Mathematical Soc.
Page : 213 pages
File Size : 43,6 Mb
Release : 2000
Category : Dirac equation
ISBN : 9780821820551

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Dirac Operators in Riemannian Geometry by Thomas Friedrich Pdf

For a Riemannian manifold M, the geometry, topology and analysis are interrelated in ways that have become widely explored in modern mathematics. Bounds on the curvature can have significant implications for the topology of the manifold. The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin structures, one obtains further information from equations involving Dirac operators and spinor fields. In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten invariants. In this text, Friedrich examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and $\textrm{spin}mathbb{C}$ structures. With this foundation established, the Dirac operator is defined and studied, with special attention to the cases of Hermitian manifolds and symmetric spaces. Then, certain analytic properties are established, including self-adjointness and the Fredholm property. An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on M lead to results about whether M is an Einstein manifold or conformally equivalent to one. Finally, in an appendix, Friedrich gives a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds. There is also an appendix reviewing principal bundles and connections. This detailed book with elegant proofs is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas. This edition is translated from the German edition published by Vieweg Verlag.

Author : Stefan Heinrich,Bernhard Banaschewski,Colin Bennett,Jacek Komorowski,Maciej Skwarczyński,Robert E. Reed,Rolando Chuaqui
Publisher : Unknown
Page : 55 pages
File Size : 44,6 Mb
Release : 1980
Category : Algebra, Boolean
ISBN : 8301011076

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A Minorization of the First Positive Eigenvalue of the Scalar Laplacian on a Compact Riemannian Manifold by Stefan Heinrich,Bernhard Banaschewski,Colin Bennett,Jacek Komorowski,Maciej Skwarczyński,Robert E. Reed,Rolando Chuaqui Pdf